Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,4,7)(17,18,22,23)(19,24,25,20)(21,26,29,30)(27,31,32,28)(34,38)(36,44)(39,41)(42,48);(10,13)(14,16)(18,23)(19,25)(21,29)(28,31)(33,34,37,38)(35,41,43,39)(36,45,44,40)(42,46,48,47);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,2)(3,6)(4,7)(5,8)(9,11,12,15)(10,14,13,16)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,4,5,3)(2,7,8,6)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fgi,d^2=h,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1688130);